The invention relates to the field of puzzles and in particular to a mathematical puzzle in form of a cube with rotatable sections, whose goal is to rotate the pieces of the cubic puzzle so as to complete a xe2x80x9cMagic Squarexe2x80x9d on each of the six faces of the puzzle.
The physical components of the puzzle include a cubic rotatable block type of puzzle that in its simplest form (a 3xc3x973 array on each of the six faces) would be comprised of 27 individual sections. 26 of the sections are visible and the 27th is the core section in the center of the puzzle upon which the other sections rotate. Other means to achieve the rotation of the core section may be used without varying from the spirit of the invention.
This underlying structure may be described by reference to the RUBIK""S CUBE, (trademarked name for cubic puzzle) which would have the same number of rotating sections (26) that this puzzle would have in its simplest form. Moreover, the underlying physical construction of a center section in connection with the peripheral sections suggests that this sort of construction can be used in the present invention. There may be more sections (for higher order arrays, e.g. 4xc3x974 and 5xc3x975) but the underlying principle of a central section in connection with the peripheral sections that allow sections of the puzzle to move together would be established.
Each face of the puzzle is divided into 9 sections; in the case of Rubik""s Cube, the 9 sections are supposed to all be of the same color when the puzzle is solved correctly. In the case of the invention herein, the 9 sections will form a xe2x80x9cmagic cubexe2x80x9d when the sections are correctly lined up which means that all of the cells in the array on each of the six faces will add up to the same number. The cells will add up orthogonally (up/down or left/right) as well as diagonally xe2x80x9cin spacexe2x80x9d, see below.
It is first necessary to define what is meant by a magic cube for purposes of this invention and to determine what will constitute completing the puzzle.
W. S. Andrews first defined a magic square as a series of numbers so arranged in a square that the sum of each row and column and of both the corner diagonals shall by the same amount which may be termed the summation. (W. S. Andrews, Magic Squares and Cubes, 2nd edition, Dover Publication Inc. New York, 1917, p 1)
Martin Gardner defined a standard magic square as;
xe2x80x9c . . . a square array of positive integers from 1 through N2 arranged so that the sum of every row, every column, and each of the two main diagonals is the same. N is the xe2x80x9corderxe2x80x9d of the square. It is easy to see that the magic constant is the sum of all the numbers divided by N. The formula is;
(1+2+3 . . . +N2)/N=xc2xd(N3+N)
xe2x80x83The trivial square of order 1 is simply the number 1 and of course it is unique. It is equally trivial to prove that no order-2 square is possiblexe2x80x9d (Martin Gardner, Time Travel and Other Mathematical Bewilderments, W H Freeman and Co. New York, 1988, p 214.
By way of illustration; FIG. 1 shows a 3xc3x973 magic square. The sum of the numbers of any row column or diagonal (a diagonal drawn through the center cell) adds up to 15. Note also that the sum of any two opposite numbers (e.g. 3 and 7 in this example) in the magic square is 10 which is twice that of the center number (5 in this case) or N2+1.
FIG. 2 shows an example of a 4xc3x974 magic square where the sum of any column, row or corner diagonal is 34. The sum of two opposite numbers is 17 which is the sum of the first number (1) and last number (16) of the series in this case.
FIG. 3 shows a 5xc3x975 magic square. Again with the same properties when the columns, rows and diagonals are added up. The sum of two opposite numbers is twice that of the center number or n2+1.
As it turns out however, such magic squares do not exist that meet, precisely, these requirements when we move to three dimensional arrays. I.e. arrays that are arranged in space so that one can sum them up along different dimensions.
Again Gardner
xe2x80x9cIt is natural to extend the concept of magic squares to three dimension and even higher ones. A perfect magic cube is a cubical array of positive integers from 1 to N3 such that every straight line of N cells adds up to a constant. These lines include the orthogonal and two main diagonals of every orthogonal cross section and the four space diagonals. The constant is;
(1+2+3 . . . +N3)/N2=xc2xd(N4+N)
xe2x80x83xe2x80x9cThere is of course, a unique perfect cube of order 1 and it is trivially true that there is none of order 2. Is there one of order 3? Unfortunately, 3 does not quite make it . . . Annoyed by the refusal of such a cube to exists, magic cube buffs have relaxed the requirements to define a species of semi-perfect cube that apparently does exist in all orders higher than 2. These are cubes where only the orthogonals and four space diagonals are magic. Let us call them Andrews cubes since W. S. Andrews devotes two chapters to them in his pioneering Magic Squares and Cubes.xe2x80x9d (1917). The order 3 Andrews cube must be associative, with 14 in its center. There are four such cubes, not counting rotations and reflections. All are given by Andrews, although he seems not to have realized that they exhaust all basic types. (Gardner, Time Travel and Other Mathematical Bewilderments, p. 219).
It is with respect to this type of xe2x80x9cAndrews Cubexe2x80x9d that Gardner refers to that we will refer to as a xe2x80x9cMagic Cubexe2x80x9d for purposes of this invention. Note that this means that only the orthogonals (rows and columns illustrated by arrows 7/8 in FIG. 8) and the four xe2x80x9cspace diagonalsxe2x80x9d (arrow 3 in FIG. 8) meet the definition of the sums being the same. Ordinary diagonals (as one goes in a diagonal direction across the face of the cube) will not necessarily sum to the same number.
Note in contrast to xe2x80x9cordinary diagonalsxe2x80x9d that a xe2x80x9cspace diagonalxe2x80x9d means a line drawn from a corner cell through the imaginary center (note again the center section is not visible to the player) and continuing in a straight line till it reaches the corner section that is opposite from the corner we started at. See arrow 3 running through the center of the 3xc3x973xc3x973 cube in FIG. 8 and having end points in a corner cube for a total of three numerical values.
There are four such center based, xe2x80x9cspace diagonalsxe2x80x9d in a cube and in a 3xc3x973 array, this space diagonal must have 3 numbers that are summed together (just like the orthogonals in a 3xc3x973xc3x973 cube). Two of the numerical values corresponding to the corner sections of the puzzle and the other value corresponding to an imaginary center section for a total of three numberes to produce the xe2x80x9cmagic sum.xe2x80x9d (A xe2x80x9ccorner sectionxe2x80x9d means like cubic section 1 in FIGS. 6/8)
The numerical value of the xe2x80x9ccenter corexe2x80x9d section may be imagined by the user because it cannot be seen when the puzzle is in normal use, and hence, there is no need to physically put a numerical value on that piece of the puzzle.
The same sort of relation holds with respect to 4xc3x974xc3x974 and higher order arrays. In the case of a 4xc3x974xc3x974 cube, the central core that cannot be seen will form a 2xc3x972xc3x972 cube (i.e. inside the larger 4xc3x974xc3x974 cube). See FIG. 9; 5 denotes the xe2x80x9ccentral corexe2x80x9d (normally unseen by the user).
The space diagonal in the 4xc3x974xc3x974 will thus cut through the center of this 2xc3x972xc3x972 core and thereby hit two members of the smaller 2xc3x972xc3x972 cube. So the same relationship holds, as above, only this time we will use four numbers to be summed up. This is true for each space diagonal as well as the orthogonals.
The same is true for 5xc3x975xc3x975 arrays with the central core in this case being 3xc3x973xc3x973 cube that is unseen by the user. In this case, a space diagonal will start/end at the two corners of the 5xc3x975xc3x975 and three cubes from the central 3xc3x973xc3x973 core will together form another space diagonal. This time the total number of numbers to be summed in the diagonal is 5. (as it would be for an orthogonal in a 5xc3x975xc3x975)
Note that in this puzzle, all of the faces on a given smaller section (out of the 26 smaller sections that comprise a larger 3xc3x973xc3x973 cube) may be imagined as having the same number printed on each face of the smaller section. Obviously, many of the faces of the smaller sections are not seen by the user (e.g. 4 of the six faces of a side section 2 in FIGS. 6/8 cannot be seen by the user) but those that are seen must have the same number on each face in order for the puzzle to work.
In the case of those corner sections of the puzzle (shown by 1 in FIGS. 6/8) there are three faces of the smaller cube that are visible to the user. One for each face of the cube that this corner section forms a part of. Thus, each visible face of this smaller cubic section would have the same numerical designation upon it.
See for example the corner section marked with indicia xe2x80x9c18xe2x80x9d in FIG. 5 and the side section marked with indicia xe2x80x9c4xe2x80x9d in FIG. 5. All three visible faces of the corner section must have the numeric indicia xe2x80x9c19xe2x80x9d and both visible faces of the side section must have the number xe2x80x9c4xe2x80x9d for the puzzle to work.
What is very interesting is that even those cubes that form the xe2x80x9ccentral coresxe2x80x9d must necessarily have the same number on all those faces that are used to form the space diagonals that run through them.
The invention is a cubic puzzle having 6 faces and an Nxc3x97N array of cells on each face. Rotatable sections form the cells of the puzzle and the rotatable sections are in connection with a central section which cannot be seen when in normal use. Each cell of the puzzle has a numerical value associated with it such that when the puzzle is successfully completed the values of any row, column or xe2x80x9cspace diagonalxe2x80x9d will add up to the same number.
In the case of a 3xc3x973 puzzle there are 26 rotatable sections in order to comprise 6 faces, each with a 3xc3x973 array on each face. In the case of a 4xc3x974 puzzle there are 56 rotatable sections. And with corresponding numbers of sections for puzzles of higher orders. Such puzzle sections may be rotatable by means of a connection between the outer visible sections and an inner core, in the manner of Rubik""s Cube. Those sections of the puzzle that have more than one face that is visible to the user will have the same numerical value associated with each face of the section.
It is an object of the invention to provide a 3xc3x973 cubic puzzle having a total of 27 sections, of which 26 are rotatable, and where each of the rotatable sections have a numerical designation upon them so that the puzzle can be rotated so as to complete an Andrews type 3xc3x973 magic square on each of the six faces of the puzzle.
Another object of the invention to provide a cubic puzzle of an order Nxc3x97N, having rotatable sections, and each section having a numerical designation so that the puzzle can be rotated so as to complete an Andrews type Nxc3x97N magic square on each of the six faces of the puzzle.
Other objects and advantages will be seen by those skilled in the art once the invention is shown and described.